On the other hand, as it is well known, black strings in general relativity suffer from the Gregory-Laflamme instability
[1]}, [2]}.
Although this instability has a perturbative nature, it is also manifested in the comparison of the entropies
between the black hole and the black string. When the entropies are compared as a function of their masses,
there is a critical mass below which the black hole has higher entropy, and it is therefore thermally favored.
Above such critical mass the behavior is the inverse.
Here, to study the possible existence of a Gregory-Laflamme transition we will take the latter path because the study of the perturbative stability
involves linearizing and decoupling a set of very complicated equations.We hope to return to the study of the perturbative stability in a future publication.
In fact, using the equations of the previous subsection, we can compare the entropies as functions of the radius at fixed equal mass.
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